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%I #9 Jan 27 2021 22:36:44
%S 1,1,5,53,977,27649,1111429,60147205,4213400897,370767834593,
%T 40025019652901,5199763957426741,800136077306754385,
%U 143904538461745813153,29906871652295426507237,7111902097369951568209349,1918658066681198636106335489,582817397769914314847061436225
%N a(n) = (n!)^2 * Sum_{k=0..n} (-1)^(n-k) * (k+1) / ((n-k)!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - x)^2.
%p a:= n-> n!^2 * add((-1)^k*(n-k+1)/k!^2, k=0..n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jan 27 2021
%t Table[n!^2 Sum[(-1)^(n - k) (k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 17}]
%t nmax = 17; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A000255, A073701, A336809.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 27 2021
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